Generalizing the Generalized Mean-Value Theorem

Abstract Undergraduate students majoring in mathematics often face difficulties in comprehending mathematical proofs. Inspired by a number of studies related to students’ proof comprehension, and Mejia-Ramos et al.’s study in particular, a test was designed in relation to the proof comprehension of the Cauchy Generalized Mean Value Theorem (CGMVT). The test mainly focused on (a.) investigating students’ understanding of relations between the statements within the CGMVT proof and (b.) the relations between the CGMVT and other theorems. Thirty-five first-year university students voluntarily participated in this study. In addition, 10 of these students were subsequently interviewed to seek their opinion about the test. Test results indicated that most of the students lacked an understanding of the relations between the mathematical statements within the CGMVT proof, and the relations between the CGMVT and other theorems. The results of interviews showed that this type of assessment was new to students and helped them to improve their insights into mathematical proofs. The findings suggested such a test design could be used more frequently in assessments to aid instructors’ understanding of students’ proof comprehension and to teach students how mathematical proofs should be learned.

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On some mean value theorems of the differential calculus

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700047109 ◽

1971 ◽

Vol 5 (2) ◽

pp. 227-238 ◽

Cited By ~ 2

Author(s):

J.B. Diaz ◽

R. Výborný

Keyword(s):

Differential Calculus ◽

Mean Value ◽

Mean Value Theorem ◽

Mean Value Theorems ◽

Partial Converse ◽

Generalized Mean ◽

Vector Valued Functions ◽

Vector Valued ◽

Special Case

A general mean value theorem, for real valued functions, is proved. This mean value theorem contains, as a special case, the result that for any, suitably restricted, function f defined on [a, b], there always exists a number c in (a, b) such that f(c) − f(a) = f′(c)(c−a). A partial converse of the general mean value theorem is given. A similar generalized mean value theorem, for vector valued functions, is also established.

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An Error Compensation Method for Improving the Properties of a Digital Henon Map Based on the Generalized Mean Value Theorem of Differentiation

Entropy ◽

10.3390/e23121628 ◽

2021 ◽

Vol 23 (12) ◽

pp. 1628

Author(s):

Yashuang Deng ◽

Yuhui Shi

Keyword(s):

Chaotic System ◽

Error Compensation ◽

Chaotic Systems ◽

Mean Value ◽

Mean Value Theorem ◽

Two Dimensional ◽

High Complexity ◽

Digital World ◽

Higher Dimensional ◽

Generalized Mean

Continuous chaos may collapse in the digital world. This study proposes a method of error compensation for a two-dimensional digital system based on the generalized mean value theorem of differentiation that can restore the fundamental performance of chaotic systems. Different from other methods, the compensation sequence of our method comes from the chaotic system itself and can be applied to higher-dimensional digital chaotic systems. The experimental results show that the improved system is highly consistent with the real chaotic system, and it has excellent chaotic characteristics such as high complexity, randomness, and ergodicity.

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